So what is the contraceptive mandate by Obamacare and why the religious group are seeking exemptions? What is the causality and relationship...?
Obamacare regulations mandate that all employers offer health insurance that pays for certain kinds of contraception. Anti-abortion activists believe that some of these forms of contraception abort fertilized eggs. In particular, some kinds of intrauterine devices (IUDs) and "morning-after" pills. This is disputed by pro-abortion people. Regardless, some object to those forms of contraception as causing abortions. Also, nuns think that they don't need contraception, as they are celibate.
So a nunnery sued the federal government for forcing them to pay for health insurance that included contraception. Gorsuch sided with the nuns.
Does "reject liberal objections" means that he may object it but just don't want to make it clear? What is his attitude?
The first amendment prohibits the federal government from establishing a religion. The fourteenth amendment is generally interpreted as extending that to state and local governments. Some (the liberals with objections) interpret that prohibition as barring religious displays from public ground, as that could be interpreted as favoring one religion (the one included) over others. Gorsuch thinks that is going too far. I.e. he rejects that view.
I don't know his exact views on the matter. Some believe that it is enough to simply offer all religions equal opportunity to put up relevant displays. Others go further and believe that the prohibition against establishment of a religion equally prevents suppression of such religious displays.
From one of your Economist quotes:
His unwillingness to defer to bureaucracies
There is a principle called Chevron deference that basically says that courts should assume that regulatory agencies are correctly interpreting the law in any regulations that they promulgate. The argument in favor of this is that the people at the regulatory agencies are experts on the subject that they are regulating.
This is a general principle. The regulations could be environmental, workplace safety, consumer finance, corporate accounting, etc.
Gorsuch believes that it is up to the legislature to write the laws and the judiciary to interpret them. So in his opinion, judges' interpretations (including his) should overrule those of regulators.
In my opinion, the Economist is confusing things here. Chevron deference would be to Trump's advantage at the moment, as he controls the regulators. Gorsuch's view is that the executive has to enforce the laws as written. They can't reinterpret them. Perhaps the Economist is trying to argue that Gorsuch will help minimize regulations after Trump leaves office.
I'm also not convinced that Donald Trump has a "small government ethos". While Trump ran as a Republican, he has previously been involved with various other parties. In particular he has donated more money to Democrats than Republicans. His ethos seems to be related more to being anti-regulation. As a business owner, he is familiar with how regulations can be troublesome without being effective. Trump likes many aspects of big government, e.g. military and infrastructure spending.
Other Republicans do have a "small government ethos". For example, Senators Ted Cruz and Rand Paul would fit that description. Also, Speaker of the House Paul Ryan.
Incidentally, I'm guessing that the executive order to which they referred was the one where Trump established the principle that for each new regulation added, two older regulations would be removed. The idea being that making regulators choose between new and old regulations would weed out the inefficient regulations. Note that some feel that that is overoptimistic and/or oversimplistic.